Eigenvectors and Eigenvalues
In a transformation, eigenvectors are the vectors that don't change direction, but get just scaled.
The factors by which they get scaled is called eigenvalue, and each eigenvactor has an associated eigenvalue.
There can also be no eigenvectors.
Imagine we have two vectors in the basic plane:
We put the plane through the transformation:
Every other vector drifts to another direction, but not these guys.
These vectors just get scaled.
The determinant of a matrix is the product of eigenvalues.
In a 3D rotation, the axis of rotation is an eigenvector. It doesn't change direction.
Formal definition
The eigen vectors are such that when receiving a linear transformation, they stay on the same original direction, only scaled by their eigenvalue(
Eigenspaces
How to compute Eigenvalues
How to compute Eigenvectors
This process must be done for each eigenvalue separately.
We input
We are left with a system of linear equations that we can solve using any method we want.
Examples with 3x3 matrices
Eigenvalues
Eigenvectors
Now we solve the system:
Now we assign
